A Characterization of Linearizability for Holomorphic ℂ*-Actions

Abstract

Abstract Let $G$ be a reductive complex Lie group acting holomorphically on $X=\mathbb{C}^n$. The (holomorphic) Linearization Problem asks if there is a holomorphic change of coordinates on $\mathbb{C}^n$ such that the $G$-action becomes linear. Equivalently, is there a $G$-equivariant biholomorphism $\Phi \colon X\to V$ where $V$ is a $G$-module? There is an intrinsic stratification of the categorical quotient $X/\!\!/G$, called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of $G$. Suppose that there is a $\Phi $ as above. Then $\Phi $ induces a biholomorphism ${\varphi }\colon X/\!\!/G\to V/\!\!/G$ that is stratified, that is, the stratum of $X/\!\!/G$ with a given label is sent isomorphically to the stratum of $V/\!\!/G$ with the same label. The counterexamples to the Linearization Problem construct an action of $G$ such that $X/\!\!/G$ is not stratified biholomorphic to any $V/\!\!/G$. Our main theorem shows that, for a reductive group $G$ with $\dim G\leq 1$, the existence of a stratified biholomorphism of $X/\!\!/G$ to some $V/\!\!/G$ is not only necessary but also sufficient for linearization. In fact, we do not have to assume that $X$ is biholomorphic to $\mathbb{C}^n$, only that $X$ is a Stein manifold.